Transitional pathway to elastic turbulence in torsional, parallel-plate flow of a polymer solution

doi:10.1017/S0022112006009426 Printed in the United Kingdom

Transitional pathway to elastic turbulence intorsional, parallel-plate flow of a polymer

solution

By BRUCE A. SCHIAMBERG1, LAURA T. SHEREDA1,HUA HU2 AND RONALD G. LARSON1†

1Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, USA2Proctor & Gamble, West Chester, OH 45069, USA

(Received 21 July 2005 and in revised form 12 February 2006)

Multiple scenarios have been discovered by which laminar flow undergoes a transitionto turbulence in Newtonian fluids. Here we show in non-Newtonian fluids a transitionsequence to ‘elastic turbulence’ due to elasticity from polymers, with negligible inertia.Multiple dynamic states are found linking the base flow to ‘elastic turbulence’ in theflow between a rotating and stationary disk, including circular and spiral rolls. Also,a surprising progression from apparently ‘chaotic’ flow to periodic flow and then to‘elastic turbulence’ is found. These transitions are found in experiments where eithershear stress or shear rate is incrementally increased and then held at fixed values;the modes found following stable base flow are ‘stationary ring’, ‘competing spirals’,‘multi-spiral chaotic’ and ‘spiral bursting’ modes, followed then by ‘elastic turbulence’.Each mode has a distinct rheological signature, and accompanying imaging of thesecondary-flow field (simultaneous with rheological measurement) reveals kinematicstructures including stationary and time-dependent rolls. The time-dependent changesin the secondary-flow structure can be related to the time-dependent viscosity in thecase of several of the modes. Finally, the effect of polymer concentration on thetransitional pathway modes is studied systematically.

1. IntroductionOver the last century, various mechanisms have been proposed to describe the

emergence of a multiplicity of modes and fully turbulent flow from initially simpleflows, including stochastic or strange attractors and period doubling (Landau &Lifshitz 1995). For inertially driven flow instabilities, the transitional states leadingfrom laminar to turbulent flow in simple Newtonian fluids have been studied inidealized flow geometries such as the Taylor–Couette flow between two concentriccylinders (Swinney & Gollub 1985) and in the torsional parallel-plate flow between twocoaxial parallel disks (Shouveiler, Le Gal & Chauve 2001; Cros & Le Gal 2002). Forthese fluids, in flows with circular streamlines, such as the Taylor–Couette example, thecentrifugal force can lead to a succession of bifurcations with increasing flow rate (orReynolds number Re), producing initially simple cellular rolls, followed by spatiallyand/or temporally modulated rolls, quasi-periodic flow, and chaotic or turbulent flows(Swinney & Gollub 1985). In the related parallel-plate circular-streamline geometry,

† Author to whom correspondence should be addressed: rlarson@umich.edu

192 B. A. Schiamberg, L. T. Shereda, H. Hu and R. G. Larson

experimental studies show a similar progression, including formation of circular andspiral rolls, and turbulent structures, as well as possible spatiotemporal intermittency(Shouveiler et al. 2001; Cros & Le Gal 2002). Here we show experimental observationsof a transitional pathway to turbulence, driven by a completely different non-inertialforce, namely fluid elasticity. We find that the hitherto unexplored transitional pathwayto ‘elastic turbulence’ shows features analogous to those leading to inertial turbulence,inviting efforts to discover theoretical reasons for these similarities.

It has been known for at least two decades that trace amounts of highly elasticpolymer in non-Newtonian viscoelastic solutions (with negligible inertia) can lead topurely elastic flow instabilities (Nguyen & Boger 1979; Larson, Shaqfeh & Muller1990; Larson 1992; Shaqfeh 1996). Because polymeric fluids are often more viscousthan simple Newtonian liquids, inertial instabilities are suppressed, but instabilities andsecondary flows can occur in polymeric fluids, even at vanishingly small Re, as a resultof nonlinearities associated with the elasticity of the fluid (Nguyen & Boger 1979;Larson et al. 1990; Larson 1992; Shaqfeh 1996). The clearest and simplest examplesof elastic instabilities occur in the Taylor–Couette flow and the torsional rotatingparallel-disk shear flow, of viscous highly elastic fluids, where experiments and theoryclearly demonstrate a mechanism by which elastic ‘normal stresses’ give rise tobifurcations to secondary flows (Phan-Thien 1983; Jackson, Walters & Williams 1984;Magda & Larson 1988; Larson et al. 1990; McKinley et al. 1991; Byars et al. 1994).These ‘normal stresses’ are produced by polymer molecules stretched along circularstreamlines, which squeeze fluid inward, leading to a radial pressure gradient (withpressure highest at the inner cylinder), and producing for example the famous ‘rodclimbing’ effect first explained by Weissenberg (1947). (In inertial flows, an analogouscentrifugally-driven pressure drop can occur, with pressure greatest at the outercylinder.) The magnitude of polymer stretch is controlled by the Weissenberg number:

Wi ≡ γ λ (1)

where γ is the shear rate and λ the characteristic polymer relaxation time. Theanalogy between inertial and elastic secondary flows extends further; at high enoughWeissenberg number (though still with negligible inertia) the elastic parallel disk flowscan show a highly irregular state with spatial and temporal characteristics similarto those of turbulence (Groisman & Steinberg 2000). What remains unexplored,however, is the sequence of bifurcations in a viscoelastic fluid that transforms simpleshearing flow to ‘elastic turbulence’ at high Wi, and how this new route compares tocounterpart transitions, such as the inertial turbulence cascade.

In non-Newtonian polymeric fluids, because polymer molecule extension is depen-dent on local velocity gradient (Larson 1999), unstable modes (which alter the localvelocity gradient relative to base flow) can dramatically change the rheologicalresponse of the fluid, such as apparent viscosity and first normal stress difference.In these experiments, both imaging of the secondary-flow kinematic structure andmeasurement of associated, time-dependent viscosity are performed. Measurementsof the time-dependent apparent viscosity provide distinct rheological signatures ofthe various modes.

2. Prior experimental studies of instabilities in torsional shearing flowHere we summarize experimental findings on the two limiting cases for the torsional

parallel-plate flow: the initial mild regular elastic instabilities (Magda & Larson 1988;McKinley et al. 1991; Byars et al. 1994) and the highly irregular ‘elastic turbulent’

Transitional pathway to elastic turbulence 193

state itself (Groisman & Steinberg 2000, 2004). Then we describe recent observationsof inertial instabilities and the transition to turbulence in torsional parallel-plate flowof a Newtonian flow.

2.1. Onset of elastic instability

Magda & Larson (1988) showed in an experimental study that the apparent ‘anti-thixotropic’ (or shear thickening) transitions observed earlier (Jackson et al. 1984)were, instead, due to instabilities that occurred at a critical shear rate with an inversedependence on gap, d , in the torsional parallel-plate flow (Magda & Larson 1988):

γc ∝ 1/d. (2)

The gap range studied was d/R = 0.024–0.16 (where R is disk radius) for a 1000 p.p.m.polyisobutylene (PIB) fluid (molecular weight Mw = 2.7 × 106 g mol−1, Oldroyd-Brelaxation time of 10 s). Furthermore, the role of molecular weight was examined (inthis case using a cone-and-plate geometry), demonstrating that the critical shear rateis dependent on molecular weight, and for the lowest molecular weight solution tested,no instability was observed (test solutions were all 1000 p.p.m. PIB, with molecularweights of 1.3 × 106, 2.7 × 106 and 4 × 106–6 × 106 g mol−1, and Oldroyd-B relaxationtimes of 2, 10 and 27 s, respectively).

McKinley et al. (1991) directly imaged an instability pattern and observed that itincluded three-dimensional radially stratified and time-dependent secondary flowstructures for a 3100 p.p.m. PIB solution (Mw = 1.8 × 106 g mol−1, Oldroyd-B relaxa-tion time of 0.794 s). The instability was found to occur at a critical rotational rateor Deborah number (here Deborah number is defined as De ≡ λΩ , where Ω is theangular velocity), consistent with Magda & Larson’s (1988) results (equation (2)).Multiple gap-to-radius ratios were studied, from 0.027 to 0.16, and, in imaging experi-ments, both axisymmetric and non-axisymmetric modes appeared in the flow; therewas evidence for an initial overstable (propagating) axisymmetric mode in an experi-ment at d/R = 0.127. These workers found that multiple shear stress states exist for agiven shear rate for the parallel-plate flow transition, in a series of experiments wherethe shear rate was initially above the critical condition and then reduced to find theshear rate at which the flow is no longer unstable; these results showed hysteresis andwere reported for d/R = 0.14.

Byars et al. (1994) directly imaged and quantified the most dangerous instabilitymode’s wave speed and wave dimensions by analysing a time series of images, for twoPIB fluids (3100 p.p.m., Mw = 1.8 × 106 g mol−1, 0.794 s Oldroyd-B relaxation time; and2000 p.p.m., Mw = 1.8 × 106 g mol−1, 1.24 s Oldroyd-B relaxation time) and two aspectratios (d/R = 0.05 and 0.088). The azimuthal wavenumber m, radial wave numberα, and wave speed (defined as the imaginary part of κ divided by α, where κ is acomplex variable which contains information about instability growth rate and timedependence) of the normal mode equations used to model the linear stability of theflow were determined for these fluids and aspect ratios; these normal mode equationsfor velocity and stress perturbations in the torsional parallel-plate geometry are ofthe form (Byars et al. 1994):

F (r, θ, z, t) = f (z) exp[iαr + imθ + κt], r = −m/α(θ + θo). (3)

The value for m must be an integer, α > 0, m/α is the ‘winding number’, and r, θ andz are the standard cylindrical coordinates. In experiments where the rotational rateof the shearing plate was gradually increased up to and beyond the critical condition,

Byars et al. observed initial overstable secondary-flow modes that were either axisym-metric (m =0) or non-axisymmetric (m = 0), depending on the fluid rheology andaspect ratio d/R. Additionally, at the same shear rate where the initial mode wasobserved in the flow, it was observed that the unstable flow (as a function of time) cantransition to an aperiodic state. They also showed that the secondary flow field (sub-sequent to formation of the initial unstable mode) possessed wavelengths shorter thanthe initial mode’s wavelength. Finally, the observed initial secondary flow occurredonly in an annular region (beyond a first critical radius and less than a second criticalradius); this experimental result was explained by an accompanying linear stabilityanalysis using a constitutive equation that included shear thinning which predictedthe disappearance of the instability beyond a second critical radius.

Recent theoretical advances in understanding elastic effects in torsional parallel-diskshear flow include work by Avagliano & Phan-Thien (1996, 1999) on the effects of afinite domain and more realistic rheological models on elastic instabilities (althoughtheir work is limited to linear stability and axisymmetric perturbations). Additionally,Olagunju (1994) has studied how capillary, inertial and gravitational forces in thebase flow of a torsional parallel-plate flow with a free radial surface, entail weaksecondary flow near plate edges, as opposed to purely circumferential flow.

2.2. Elastic turbulence

Groisman & Steinberg (2000) showed that many of the principal features of inertialturbulence are present in a highly disordered viscoelastic flow, which they named‘elastic turbulence’ (test solution was 80 p.p.m. polyacrylamide, Mw = 18 × 106; 3.4 srelaxation time based on oscillatory measurements). Both imaging of secondary flowand Doppler velocimeter measurements in a torsional parallel-plate flow showed apower law decay over about a decade in the flow’s wavelength (scaled with the gap)and frequency, respectively. Measurement of solution shear stress values for the ‘elasticturbulent’ state showed an increase by nearly a factor of 20 (depending on the valueof d/R) relative to a hypothetical laminar flow (Groisman & Steinberg 2000). Theyalso showed (2004) that ‘elastic turbulence’ was accompanied by significant increasesin the rates of mass transfer relative to what would be expected from simple diffusionin the corresponding laminar flow.

In the original characterization of the ‘elastic turbulent’ state by Groisman &Steinberg, the large gap ratio (d/R = 0.263 and 0.526) in the torsional parallel-plateflow (where an outer fluid retaining wall is necessary because of the large gap, andR here is the upper plate radius) and polymer solution properties evidently result ina more-or-less direct transition from the base flow to ‘elastic turbulence,’ although asingle mediating structure with the shape of a ‘toroidal vortex’ is noted (Groisman &Steinberg 2000, 2004).

2.3. Inertial transition to turbulence

Finally, we summarize here recent Newtonian-flow experiments by Schouveiler et al.(2001) on the inertial transitions leading to turbulence in torsional parallel-plate flow,for a range of aspect ratios d/R as a function of increasing Reynolds number, ReR

(where ReR ≡ ΩR2/ν and ν is the kinematic viscosity. Here, R refers to the stationarydisk radius). In their large-gap experiments (d/R = 0.0714–0.1429), the first unstablemode to form from the basic flow is inward-propagating axisymmetric rolls whichform near the edge of the flow field, followed at higher ReR by formation of outward-propagating spiral rolls near the edge of the flow field (with the axisymmetric modestill present in the inner region). In this gap range, further increasing ReR resultsin ‘wave turbulence.’ For the small-gap experiments (d/R = 0.0071–0.0179), the first

unstable mode observed consists of spiral rolls which are either inward- or outward-propagating. A further increase in ReR in this gap range leads to ‘solitary waves’,which are spiral turbulent structures that are not periodic with respect to space ortime; these ‘solitary waves’ co-exist with and overlie the spiral rolls from the previousmode. Further increase of ReR for specific values of the aspect ratio within the small-gap range can lead to the formation of ‘spots’ near the edge of the flow field; these‘spots’ spiral inward, ‘passing through’ the spiral roll and ‘solitary wave’ structures.Finally, upon still further increase in ReR , the spiral and ‘solitary wave’ structures arenot observed; rather, ‘spots’ are the only structure observed (now migrating to radialpositions closer to the flow centre) and the flow becomes turbulent in the outer radialregion. At intermediate values of the aspect ratio (d/R =0.0179–0.0714), Schouveileret al. observed not only some of the modes seen at the other aspect ratios, but alsostationary spiral rolls, which were not seen at the other gap ratios. The test fluid inthese experiments was water.

In what follows, we report results of experiments with elastic polymeric fluids atsmall Reynolds number in the torsional parallel-plate flow with a smaller gap ratiothan was used by Groisman & Steinberg, where we find a rich series of flow statesconnecting the base flow to ‘elastic turbulence’, each possessing distinct time-dependentrheological signatures.

3. Experimental methodsWe study polyacrylamide solutions in torsional parallel-plate shearing flow, visualiz-

ing secondary flow-field states and simultaneously measuring solution materialproperties using a torsional rheometer in a ‘stress-controlled mode,’ where characteri-stic shear stress, σ (defined below) is incrementally increased, or in a ‘rate-controlledmode’, where the shear rate γ is incrementally increased (as γ varies linearly withradial position, we report its value at the outer radius R). Secondary flow is imagedby means of light-reflective tracer particles in a custom-designed shearing cell (withtransparent bottom fixture) that is fixed to the rheometer base (see figure 1). Thefluid is confined by a cup, the bottom of which serves as the plate. The gap ratio d/R

was set to d/R = 0.204, where d is the gap and R is the radius of the upper plate.In a stress-controlled mode, an increase in apparent viscosity η(γ ) (or the formationor growth in amplitude of an unstable mode) signals the rheometer to decrease γ

relative to experiments where γ is the control parameter and large increases in η(γ )occur:

σ (γ ) = η(γ )γ . (4)

Our classification of states is based on secondary flow-field imaging, rheologicalchange, and fast fourier transforms (FFTs) of time-dependent rheology. Becausefeatures of one mode often persist or reappear in subsequent modes (since transitionsbetween modes are gradual in some cases), we report the dominant flow modesand general trends observed, and first illustrate the modes using a representativeexperiment of a single fluid. To characterize observations of complex waveformpatterns or ‘roll cells’, the normal mode equations from Byars et al. (1994; equation (3))are used; in describing the different spiral patterns observed in our experiments, weassume α is constant and define Wn ≡ m/α. Finally, we note that in our so-called‘controlled σ ’ experiments, the ‘shear stress’ we report is really the applied torque,T , converted to units of σ (note that the rheometer actually controls torque). Theequation used by our rheometer software to apply a ‘shear stress’ value is based on

σ(t), γ(t)

Camera

Figure 1. Flow-visualization apparatus. The dimensions are R = 12.5 mm and R2 = 15.5 mm.The fluid meniscus and free surface are represented by the lines connecting the upper shearingplate to the outer wall. The flow is visualized by seeding test fluid with reflective tracerparticle-flakes, which align in the flow direction. Regions of high reflected light intensitycorrespond to flow in the (r, θ )-plane, and regions of low reflected light intensity correspondto z-direction flow.

the relationship σ = 2T/πR3 between T and σ (at R) in a Newtonian fluid withoutsecondary flow. Since, in our case, the flow that actually occurs is far from meetingthese conditions, σ should not be thought of as an actual shear stress (although wewill for simplicity call it ‘stress’, ‘shear stress’ or σ throughout this paper), but ratheras the torque, expressed in units of stress. In addition, given that our set-up requiresa bottom cup, which serves as the bottom plate, there will be additional discrepancybetween applied torque and the actual imposed shear stress at a location in theshearing apparatus. Similarly, the ‘controlled shear rate’ experiments are conductedat a controlled rotation rate of the upper plate, expressed as the shear rate at therim, for a hypothetical fluid with no secondary flow at the rim (we will refer to thisrim shear rate as ‘shear rate’ or γ throughout this paper). Finally, we will report anapparent ‘viscosity’ or η throughout this report which is merely the ‘stress’ definedabove divided by the apparent ‘shear rate’ at the rim. Of course, when secondary flowis occurring, this ‘viscosity’ indicates the overall level of viscoelastic stress generatedby the flow and is not at all a material property of the fluid.

In the device depicted in figure 1, a mirror was placed at a 45 degree tilt underthe bottom glass window, and diffuse illumination was provided using a fibre opticlight source, pointed in the direction of (but not directly at) the sample. Reflectivetracer seed particles in the test solution are mica/titanium dioxide at 0.1 g/100 ml(Mearlin r© Superfine, Engelhard, NJ). The bottom surface of the upper shearing platewas coated with black ink to improve contrast. Plates are aligned parallel within5 µm over a distance of at least 35 mm, and plates are adjusted to be coaxial usingmachined semi-cylindrical plastic slabs which fit into the annular region betweenupper plate edge and inner cylinder wall. The bottom plate is a quartz window, with

anti-reflective coating (Esco Products, Oak Ridge, NJ) with a precision bore glass tubesegment affixed (Wilmad Labglass, Buena, NJ). The rheometer is a TA InstrumentsAR1000. Fluid temperature is monitored before and after each experiment and thesolution is allowed to equilibrate for a 20 min (minimum) period before the start ofshearing. Stress is incremented in different step sizes, in an attempt to probe statesand transitions better.

Flow-field images were recorded in analogue format during experiments lastingaround 8 h. For each experiment, images were converted to digital and then greyscaleformat, and, for the purposes of clarifying illustration, multiple images over anarrow time window (between 0.1 and 0.33 s, depending on the mode) were averagedseparately for each mode (this averaging alters slightly ‘spiral bursting’ and ‘elasticturbulent’ image patterns, but the images shown here represent the patterns well);then the base flow image was subtracted from the images for all other modes for theparticular experiment. After this, the flow-field region was cropped from the largerimages. Finally, the contrast and brightness of each image was adjusted to the samedegree with reference to each experiment.

4. Fluid characterization and preparationOur initial test solution is comprised of 492 p.p.m. polyacrylamide of molecular

weight 18 × 106 gmol−1 (Polysciences, Warrington, PA) dissolved in a viscosifiedaqueous solution of 65.9 % sugar (sucrose), 0.98 % sodium chloride (NaCl), in water.This polymer concentration is slightly above the overlap concentration, just into thesemi-dilute regime. The overlap concentration, c∗, is defined by c∗ ≡ ν∗Mw/NA, whereν∗ is the number of polymer molecules per unit volume necessary to yield one polymermolecule, on average, in a fluid volume of R3

g (Rg is the polymer radius of gyration),where Mw is the polymer molecular weight, and NA Avogadro’s number. Becauseexcluded volume interactions are of unknown significance, and affect Rg , we hereestimate c∗ empirically, as the concentration at which addition of polymer roughlydoubles the solution viscosity η0 (where η0 = ηp,0 +ηs) relative to the solvent viscosityηs , such that η0 ∼ 2ηs (Larson 1999).

For our study of the effect of polymer concentration on the transitional pathway(and verification of the transitional pathway using fluids with different rheologicalproperties), polymer test solutions are comprised of 200, 400 and 800 p.p.m. poly-acrylamide of molecular weight 5 × 106–6 × 106 g mol−1 (Acros Organics, NJ), alsodissolved in a viscous syrup of 65.9 % sugar (sucrose), 0.98 % NaCl, in water.The 400 p.p.m. solution appears to be slightly above the overlap concentration (theborderline between dilute and semi-dilute regimes), based on the empirical estimatedescribed above.

In cone-and-plate experiments, we determined the steady-state relaxation time, λss ,based on the Oldroyd-B relation λss = Ψ1/2ηp,0 (Larson et al. 1990), for each polymersolution (see table 1). For the 492 p.p.m. fluid (Mw =18 × 106 g mol−1), λss was esti-mated in this way as 1.0 ± 0.01 s. In figure 2, we show rheometric measurements of the5–6 × 106 g mol−1 solutions, from which we obtain estimates for ηp,0 and Ψ1,0. Basedon these measurements, we estimate steady-state relaxation times, λss , as 2.3 ± 0.08,1.5 ± 0.02 and 1.7 ± 0.04 s for the 800, 400 and 200 p.p.m. solutions, respectively.We use λss to calculate Wi in this report. In table 1, we summarize the rheologicalproperties of the different fluids (where β ≡ ηs/η0). We encountered some difficultyin observing the zero shear viscosity, η0, of the 18 × 106 g mol−1 fluid, as it had anunusually high viscosity at very low shear rates (either due to instrument sensitivity

800 p.p.m. 400 p.p.m. 200 p.p.m. 492 p.p.m.polyacrylamide polyacrylamide polyacrylamide polyacrylamide(Mw = 5–6 × (Mw = 5–6 × (Mw = 5–6 × (Mw = 18 ×106 gmol−1) 106 gmol−1) 106 gmol−1) 106 gmol−1)

η0 (Pa s) 1.49 0.68 0.43 0.95ηp,0 (Pa s) 1.24 0.43 0.18 0.70ηs (Pa s) 0.25 0.25 0.25 0.25β ≡ ηs/η0 0.17 0.37 0.58 0.26Ψ1,0 (Pa s2) 5.8 ± 0.19 1.3 ± 0.01 0.6 ± 0.01 1.5 ± 0.02λss (s) 2.3 ± 0.08 1.5 ± 0.02 1.7 ± 0.04 1.0 ± 0.01

Table 1. Rheological properties of solutions, at 19 C.

800 p.p.m.

400 p.p.m.

200 p.p.m.

100 10110–1

10–1

100 101 102

Shear rate (s–1)

Figure 2. Fluid characterization. Plots of η, ηs , and N1 vs. γ for the 800 p.p.m., 400 p.p.m. and200 p.p.m. polyacrylamide (Mw = 5–6 × 106 g mol−1) solutions. The inset shows plots of Ψ1 vs.γ for the three fluids. The measurements were performed in using a cone-and-plate fixture at19C.

limitations or possible solution gel effects at low shear rates), and estimated η0 whenthe viscosity began a rough plateau, starting at around 1.8 s−1. Finally, we add thatfor a similar polyacrylamide (Mw = 18 × 106 g mol−1) fluid (not included in this study),we measured a relaxation time from the decay of N1 following cessation of shear,finding this relaxation time to be about 4 times the Oldroyd-B relaxation time; thisratio is consistent with what has been observed for other Boger fluids, where the ratiowas approximately 2.4 to 3 (Shaqfeh, Muller & Larson 1992).

10–1 100 101 102

Shear stress (Pa)

Shear rate (s–1)

10–1 100 101

Figure 3. Elastic instability progression, general rheological trends for 492 p.p.m. polyacryl-amide solution (Mw = 18 × 106 g mol−1). Plotted is ηavg vs. σ over the range 0.16–54.9 Pa, corres-ponding to Wi= 0.15–3.9 and Re= 0.0014–0.036. The inset shows a plot of ηavg vs. γavg. (Note:to calculate Wi and Re for unstable modes, the average shear rate for the constant stress stepis used.)

We note also that in the cone-and-plate measurements shown in figure 2, thereis clear evidence of elastic instabilities, as seen in both the apparent shear viscosityand first normal stress coefficient. Rheological measurements of test solutions wereperformed using a TA Instruments, Rheometric Series, ARES LS rheometer, withall measurements performed at 19 C using a cone-and-plate fixture (with a radiusof 50 mm, and cone angle of 0.04 rad). Solutions are prepared by initially dissolvingpolyacrylamide in water, prior to dissolution in a concentrated sucrose and watersolution, followed by addition of NaCl.

5. Results5.1. Identification of a transitional pathway

Here we focus on secondary flows that develop under slowly increasing controlledstress for a semi-dilute, 492 p.p.m. polyacrylamide (Mw = 18 × 106 g mol−1) solution ina viscous Newtonian solvent comprised of 65.9 % sugar (sucrose), 0.98 % NaCl, inwater (see figure 3), at room temperature. Similar results were obtained for other solu-tion concentrations, molecular weights and gap ratios, as will be detailed elsewhere.The flow field is visualized over a range of fixed shear stress values, σ , and γ ismeasured. (When the experiment is changed from controlled σ to controlled γ , allmodes are still seen.) In figure 3, we plot the apparent shear viscosity versus σ ,for the range 0.16 to 54.9 Pa, corresponding to Wi =0.15–3.9 and Re= 0.0014–0.036.(Re ≡ ρV d/η0, where V is the tangential velocity at R and ρ is the solution density);we observe shear-thickening in the average viscosity (time-average viscosity for eachconstant shear stress step, typically over a 20 or 10 min period), beginning at σ = 1.4 Pa(note: there is an expected initial weakly shear-thinning region). The time-dependentshear rates exhibited when each mode is accessed at specific values of σ are shown infigure 4. Figure 4(a) represents uniform azimuthal flow or base flow in this geometry.In figure 4(b), we show the ‘stationary ring’ mode, which is an axisymmetric ‘ring’

1.0(a)

0 200 400 600

(s–1

0.50 200 400 600

(s–1

1.50 200 400 600

(s–1

Time (s)

Figure 4. Elastic instability progression, initial modes. (a) Base state, σ = 0.16 Pa, Wi= 0.15;(b) ‘stationary ring’ mode, σ = 0.96 Pa, Wi= 1.1; (c) ‘competing spirals’ mode, σ = 1.8 Pa,Wi= 1.7 (note: to calculate Wi for unstable modes, the average shear rate for the constantstress step is used).

or set of rings forming in an annular segment of the flow field, accompanied by ashear rate decrease (i.e. an increase in apparent viscosity). The axisymmetric patterninitially forms at approximately the outer edge of the sample and then moves to aprogressively smaller radial position as shear stress is increased. Then, in figure 4(c),we show the ‘competing spirals’ mode, which includes formation of an indeterminatepattern of non-axisymmetric spirals and/or axisymmetric rings at the middle to outerradial positions accompanied by a pronounced decrease in shear rate. The inner ringfrom the ‘stationary ring’ mode co-exists with the unstable modes in the outer region,but the inner ring moves to progressively smaller radial positions as shear stress isincreased. Near the end of the mode (as shear stress is increased), the centre circlebegins to blur.

2.5(a)

1.50 400 800 1200

(s–1

0 0.02 0.04Pow

1.50 400 800 1200

(s–1

0 400 800 1200

(s–1

Time (s) Frequency (Hz)

Figure 5. Elastic instability progression, final modes. (a) ‘multi-spiral chaotic’ mode, σ =2.6 Pa, Wi=1.8; (b) ‘spiral bursting’ mode, σ =7.4 Pa, Wi = 1.9; (c) ‘elastic turbulent’ mode,σ = 54.9 Pa, Wi= 3.9 (note: the scales on the ordinates of figures 4 and 5 vary). The timeperiod in these steps was 1200 s, with one measurement taken per second. The FFTs are basedon the final 1024 data points, to exclude start-up effects. To calculate Wi for unstable modes,the average shear rate for the constant stress step is used. Video sequences of these modes areavailable as a supplement to the on-line version of this paper.

In figure 5(a), we show the ‘multi-spiral chaotic’ mode, which includes an irregulartime-dependent shear viscosity, and formation of spiral modes at all radial locations.At the start of ‘multi-spiral chaos’, the ring from the competing spirals mode separatesfrom the outer region, and dark thin annular regions form on either side of the ring asthe ring grows in amplitude (shown in figure 5a). The increased-amplitude ring is thenreplaced by spirals at the same radial location (not shown). It appears that when theinner ring from the ‘competing spirals’ mode has decreased in radius and the mode isno longer stable at any radial position, the ring breaks symmetry with respect to theazimuthal coordinate, and a state of ‘chaotic’ multi-spiral modes ensues. Eventually,at fixed shear stress, the circular ring reforms and stabilizes, and the above sequenceis repeated. The pattern-formation cycle is temporally aperiodic, as is reflected in therange of excited low frequencies in η(t), revealed by the FFT (see figure 5a). Thecycle gradually degenerates as shear stress increases, and a transient low windingnumber (Wn) spiral pattern replaces the ‘ring’; and eventually, as stress is increased,the cycle begins to resemble the next mode. In figure 5(b), we show the ‘spiralbursting’ mode, which includes formation of a rapidly propagating spiral of high Wnwhich begins near the centre of the flow and propagates outward. A lesser Wn andslower-moving spiral then appears and gradually fills in the rest of the flow field,typically accompanied by a maximum in viscosity. The spirals then dampen before

the cycle repeats. The shear rate fluctuations are characterized by growth of a singledominant peak in the power spectrum which corresponds to a ‘spiral bursting’ cycle.

Finally, we show the ‘elastic turbulence’ mode in figure 5(c). We take the name ofthis mode from Groisman & Steinberg (2000) who first imaged and named a similarstructure with accompanying apparent viscosity increase similar to that in figure 5(c)(note: based on the average shear rate in figure 5(c), the apparent viscosity increase isabout 13 times the base flow viscosity; also see the final data point shown in figure 3).‘Elastic turbulence’ is a flow field characterized by very high Wn spirals or swirls,periods of loss of alignment of the centre of the secondary flow pattern with the axis(r = 0), and occasional and few ‘bursts’ or returns to the ‘spiral bursting’ mode. Theshear rate fluctuations are characterized by a broad range of excited time-dependentmodes, which appear, collectively, aperiodic. The shear viscosity shows a large increase(more than a decade) on entry into the ‘elastic turbulent’ state. The most convincingevidence that this state deserves the name ‘turbulence’ lies in the power-law spectrumof velocity fluctuations and in length scales observed, respectively, through laser-Doppler velocimetry and image analysis of mica-flake reflection patterns, carriedout by Groisman & Steinberg (2000). We have not replicated these experiments,since thorough characterization of this state to confirm the findings of Groisman &Steinberg is not the purpose of this paper. However, the appearance of the flow, ouruse of a similar fluid and similar geometry to that of Groisman & Steinberg, andour observation of a similar large increase in stress seen on entry into the flow areall convincing evidence that the flow shown in figure 5(c) is indeed the same as thatdescribed as ‘elastic turbulence’ by Groisman & Steinberg. We note that our FFTshows a broad spectrum of modes, which, however, only appears to be a power lawif binned more coarsely than done in figure 5(c). Velocimetry data, spatial patternanalysis, and extremely long runs (many hours to sample the frequency spectrumthoroughly) would be required for us to check rigorously for power-law temporal andspatial patterns.

5.2. Effect of polymer concentration

As an extension of our discovery and characterization of the modes occurringbetween base flow and ‘elastic turbulence’ in experiments with the identical polymerand molecular weight (polyacrylamide, Mw =18 × 106 g mol−1) to that first used byGroisman & Steinberg in their characterization of the elastic turbulent state itself(Groisman & Steinberg 2000, 2004), we performed experiments on new polyacrylamidesolutions with different fluid rheology, polymer molecular weight (5–6 × 106 g mol−1)and concentrations, to explore the generality of the transitional pathway and to testthe effects of systematic changes in fluid rheology. In these additional experiments weexplored the effect of changing polymer concentration – spanning the dilute to semi-dilute regimes – on features of the various modes comprising this transitional pathway,testing polymer concentrations of 200, 400 and 800 p.p.m. In these experiments, oursolvent system was also 65.9 % sugar (sucrose), 0.98 % NaCl, in water, and d/R

remained set to 0.204. We confirmed, using these new fluids, that the modes observedin the molecular weight 18 × 106 g mol−1 solution can occur in different elastic fluids,and that all previously described modes (‘competing spirals’, ‘multi-spiral chaotic’,‘spiral bursting’ and ‘elastic turbulence’) are seen in the new fluids (with few slightdifferences and exceptions in mode rheological signatures or image patterns, as will bediscussed below), at each of the three separate polymer concentrations. In addition, anew periodic mode is stabilized in the most concentrated of the three solutions tested,which we name ‘spiral oscillatory’ mode.

10–1

10–2 10–1 100 101 102

Shear stress (Pa)

10–1

10–2 10–1 100 101 102

Shear rate (s–1)

η, 800 p.p.m. ,η, 400 p.p.m. ,η, 200 p.p.m. ,

(a) (b)

Figure 6. Elastic instability progression, general rheological trends as a function of polymerconcentration for polyacrylamide with molecular weight 5–6 × 106 g mol−1. Each concentrationis tested in duplicate experiments and plotted using two different symbols for eachconcentration. (a) Plotted is ηavg vs. σ ; (b) ηavg vs. γavg for the same experiments.

In figure 6(a), we show shear stress versus apparent viscosity plots for all threesolution concentrations tested at d/R =0.204, each in duplicate. In figure 6(b), forthese same experiments, we show shear rate versus apparent viscosity (where shearrate values are average values for the fixed shear stress holding over the giventime interval). The apparent viscosity versus shear stress plots for the three solutionconcentrations roughly define a single pathway if each data set is adjusted (divided) bya factor of the ratio of zero shear viscosity of the solution to zero shear viscosity of theleast concentrated solution (see figure 7). As well, a ‘universal’ transitional pathwayslope is evident in the apparent viscosity versus shear stress plot, for the approximatelylinear (on a log–log plot), shear-thickening region of the transitional pathway from1.4 Pa to 22.9 Pa; a linear fit to this region yields a power-law slope of around 0.2 (seefigure 7; actual value of the slope is 0.202 ± 0.004). In figure 8, we show images ofthe secondary flow field for each mode (except ‘spiral oscillatory’ mode which occursbetween ‘competing spirals’ and ‘multi-spiral chaos’ in the 800 p.p.m. fluid, and isdescribed below). In figure 9, we show plots of the time-dependent shear rate foreach of the modes (excluding ‘spiral oscillatory’) at all three polymer concentrations.In figure 10, we show FFT plots of the same time-dependent shear rate data, formodes ‘multi-spiral chaotic’ to ‘elastic turbulence’. Note that for the 200 p.p.m. and400 p.p.m. polyacrylamide fluids, the long-time upward slope of the time-dependentshear rate during ‘elastic turbulence’ results in large-amplitude low-frequency peaksin the FFT (note: these peaks are off scale in the plots); we believe this long-timeupward trend in the time-dependent shear rate (corresponding to a viscosity drop)might be related to possible shear-induced polymer migration, as there is no evidenceof fluid degradation (such possible migration has been previously reported in similarviscoelastic fluids (Magda et al. 1993)).

An important difference in the responses of fluids tested is that the most concen-trated solution (800 p.p.m.) undergoes a periodic oscillatory mode which we name‘spiral oscillatory’. The ‘spiral oscillatory’ mode bears some similarity to the ‘competing

10–1

10–2 10–1 100 101 102

Shear stress (Pa)

η800 × (η200,0/η800,0): ,

η400 × (η200,0/η400,0): ,

η200 × (η200,0/η200,0): ,

Linear fit: (y = 0.202x + 0.315)

Figure 7. Elastic instability progression, a ‘universal’ transitional pathway slope. Plottedare viscosities for each of the three polyacrylamide (Mw = 5–6 × 106 g mol−1) concentrations(800 p.p.m., η800, 400 p.p.m., η400, and 200 p.p.m., η200), from duplicate experiments, where eachviscosity has been divided by the ratio of the zero shear viscosity of the solution (which wedesignate η800,0, η400,0 and η200,0) to the zero shear viscosity of the 200 p.p.m. solution, η200,0.

spirals’ mode, with the important distinction that in the former, the flow field outsidethe stationary inner ring contains a single (or few) outward-propagating spiral mode(s)that are readily observable. The time-dependent shear rate is clearly periodic, asevident in figure 11. Although only the 800 p.p.m. solutions show the distinct presenceof the ‘spiral oscillatory’ mode (no such mode is distinctly stabilized in the lessconcentrated 400 and 200 p.p.m. fluids), there is some evidence of the ‘spiral oscillatory’mode in these less concentrated solutions. Prior to the ‘multi-spiral chaotic’ mode,both the 400 and 200 p.p.m. solutions show possible ‘spiral oscillatory’-like secondaryflow (as determined from video-imaging) and periodicity in the time-dependent shearrate (as determined by FFT), suggesting that the mode may be present, possibly mixedwith other competing modes. Also, we have observed in a single experiment for the400 p.p.m. solution (after the initial onset of ‘multi-spiral chaos’), the possible presenceof ‘spiral oscillatory’ features in a hybrid ‘spiral oscillatory-multi-spiral chaotic’ state,as seen in both the time-dependent secondary-flow image patterns (regular outward-propagating spirals, plus the previously described ring-breaking cycle) and rheologicalsignatures (a strong periodic frequency, plus other low frequencies).

To determine a characteristic wave speed and wavelength of the secondary flow fieldduring the ‘spiral oscillatory’ mode, we performed image enhancement and analysis

Figure 8. Elastic instability progression for various polymer concentrations, secondaryflow images. Shown (from left to right) are the 200, 400 and 800 p.p.m. polyacrylamide(Mw = 5–6 × 106 g mol−1) fluids. Shown (from top to bottom) are base flow, ‘stationary ring’,‘competing spirals’, ‘multi-spiral chaotic’, ‘spiral bursting’ and ‘elastic turbulent’ modes. Severalof the time-dependent modes are most clearly seen in video sequences available as a supplementto the on-line version of the paper (σ and Wi for each of the modes shown above, from top tobottom, for the 200, 400, 800 p.p.m. fluid experiments are, respectively: σ = 0.094, 0.93, 1.8, 3.1,13.9 and 59.6 Pa, Wi= 0.31, 3.3, 4.7, 5.4, 7.2 and 19.4; σ = 0.096, 0.96, 1.8, 2.6, 11.9 and 69.7 Pa,Wi= 0.16, 1.9, 2.7, 2.9, 3.5 and 11.9; σ = 0.098, 0.73, 2.2, 3.7, 11.9 and 69.9 Pa, Wi= 0.12, 1.1,2.8, 2.8, 3.2 and 8.8. We note that to calculate Wi for unstable modes, the average shear ratefor the constant stress step is used).

0 200 400 600

0 200 400 600 0

0200 400 600 200 400 600

1.50 400 800 1200 0

10200 400 600 400 800 1200

1.50 400 800 1200 0

10400 800 1200 400 800 1200

20 400 800 1200 0

10400 800 1200 400 800 1200

60 400 800 1200 0

20400 800 1200 400 800 1200

Time (s) Time (s) Time (s)

(s–1

Figure 9. Elastic instability progression for various polymer concentrations, rheological signa-tures. Shown (from left to right) are the 200, 400 and 800 p.p.m. polyacrylamide (Mw = 5–6 ×106 g mol−1) fluids. Shown (from top to bottom) are base flow, ‘stationary ring’, ‘competingspirals’, ‘multi-spiral chaotic’, ‘spiral bursting’ and ‘elastic turbulent’ modes (σ and Wi for themodes shown above are the same as for the corresponding mode and fluid in figure 8, sincedata for corresponding fluids are from the same experiments).

0 0.02 0.04

0 0.02 0.04 0 0.02 0.04

0 0.02 0.04

0 0.02 0.04 0 0.02 0.04

Frequency (Hz) Frequency (Hz) Frequency (Hz)

Figure 10. Elastic instability progression for various polymer concentrations, FFTs of time-dependent rheological signatures of the ‘multi-spiral chaotic’, ‘spiral bursting’ and ‘elasticturbulent’ modes. Shown (from left to right) are the 200, 400 and 800 p.p.m. polyacrylamide(Mw = 5–6 × 106 g mol−1) fluids. Shown (from top to bottom) are ‘multi-spiral chaos’, ‘spiralbursting’ and ‘elastic turbulence’ (note that for the ‘multi-spiral chaotic’ mode for the 200 p.p.m.fluid, one of the low-frequency peaks (f ∼ 0.001Hz) is off scale, with a value of 4.5. Separately,σ and Wi for the modes shown above are the same as for the corresponding mode and fluidlisted in figure 8, since data for corresponding fluids are from the same experiments).

(s–1

0 400 800 1200Time (s)

Figure 11. Rheological signature of the ‘spiral oscillatory’ mode. Plot of the time-dependentshear rate for the 800 p.p.m. polyacrylamide (Mw = 5–6 × 106 g mol−1) fluid at σ = 3.0 Pa.

(b)(a)

–150 20 40 60

Radial position (pixels)

0:09:000:09:500:10:40

Figure 12. Determination of the characteristic wavelength and propagation frequency of the‘spiral oscillatory’ mode. (a) sample secondary flow-field image and the approximate imageregion used in the analysis (see boxed region); (b) plot of ‘image intensity profile’ vs. radial pixelposition (1 pixel= 0.067 mm) average over each of three consecutive 1 s time intervals occurring50 s apart (see main text for description of image processing, averaging and adjustments usedto obtain the ‘image intensity profile’). The image shown and data plotted here are from thesame experiment and shear stress (σ = 3.0 Pa) as shown in figure 11, where times in this figurecorrespond to 9 min, 9min 50 s and 10min 40 s after the start of a constant-shear-stress holdat 3.0 Pa. (Note: reflected light intensity is a one-dimensional average of vertical columns ofpixel intensity values in the rectangular grid for the box in (a), where larger values correspondto brighter regions. Recall that regions of high reflected light intensity correspond to flow inthe (r, θ)-plane, and regions of low reflected light intensity correspond to z-direction flow.)

of a flow-field region where the apparent time-dependent propagation of spiral rollcells was readily apparent. The secondary flow-field region was analysed over a 10 minperiod, based on image sets collected at 50 s intervals. A series of 13 digital image setsconsisting of 31 frames (each spaced 1/30 s apart), where the first image in each setoccurred, identically, 50 s apart were assembled (a set of 31 images of the base flowwas also assembled). The images were converted to black and white, and then eachset of 31 images were averaged together to produce a single enhanced image (this wasdone by averaging corresponding pixels in space over time, separately for each set of31 images). Subsequently, the base flow was subtracted from each of the enhanced‘spiral oscillatory’ mode images. A common set of pixel intensity values correspondingto the region approximated by the rectangle in figure 12(a) were used as a basis forestimating the wave speed and wavelength of the mode in the radial direction. Verticalcolumns of pixel intensity values in the rectangular grid were averaged to producea single one-dimensional profile for each of the time periods occurring 50 s apart.Separate linear fits to each intensity profile were then subtracted from each of therespective image intensity profiles. In figure 12(b), we show three successive, resultingimage intensity profiles, where the spiral roll cells are clear (note: at other timesduring the 10 min period, clear waveforms are not always present and/or two wavepeaks are not present in the spatial interval examined). Finally, the one-dimensionalprofiles for consecutive time period pairs were cross-correlated, and maximum cross-correlation values were used to determine displacements of the propagating ‘waves’between image pairs. The average ‘wave’ displacement between images examined was6.4 pixels or 0.43 mm, yielding a wave speed in the radial direction of 0.0086 mms−1.The wavelength was estimated by measuring the pixel distance between maximumintensity values for two wave peaks. As previously noted, in some cases, clear wavepeaks or wave pairs are not apparent so data was collected only from images where

two peaks were evident, and an average value was then determined from this set ofmeasurements. The average wavelength was 35.4 pixels or 2.37 mm. The wavelengthand wave speed in the radial direction yield a wave period and frequency of 276 sand 0.0036 Hz, respectively; this compares well to the frequency of 0.0039 Hz for thetime-dependent shear rate or viscosity, as seen in figure 11 (note: the FFT method for1024 points bins power into a discretized set of frequencies, including 0.0029, 0.0039and 0.0049 Hz in the low-frequency range).

The correlation between secondary-flow frequency (as determined by imageanalysis) and the frequency of the changing shear rate (or apparent solution viscosity)demonstrates that the perturbational mode frequency can be observed in rheometricmeasurements. This is probably due to the finite dimensions of the shearing apparatus.We believe this is the first demonstration of this correlation, although periodicoscillations (resulting from elastic instabilities) in the apparent viscosity of elastic fluidshave been reported elsewhere (Shaqfeh et al. 1992). Although the correlation betweenrheometric measurement and mode frequency becomes considerably more complexas the number of secondary-flow modes in a flow field increases, the connectionestablished here with a single or few modes motivates the argument that the time-dependent viscosity reflects the wavelengths and wave speeds of the unstable flowcontained within the shearing apparatus.

We find that the frequency of ‘spiral bursting’ increases progressively as a functionof decreasing polymer concentration. In figure 13, we show FFT results of the time-dependent shear rate during ‘spiral bursting’ for each concentration, taken fromduplicate experiments.

We note here the apparent differences and exceptions in the rheological signaturesor image patterns of modes at specific polyacrylamide (Mw = 5 × 106–6 × 106 g mol−1)concentrations, compared to the 18 × 106 g mol−1 molecular weight polyacrylamidesolution experiments. First, the shear rate decrease (or shear viscosity increase) oftenassociated with ‘stationary ring’ mode secondary-flow pattern can be very slight ornot detectable; although there are examples of this for each concentration, we findthis is particularly the case for the 200 p.p.m. polyacrylamide fluid (see figure 9).Additionally, although there is a dominant peak in the FFT of the time-dependentshear rate for ‘spiral bursting’ in the 400 p.p.m. fluid, these experiments also showsecondary peaks of the order of 40 to 60 % of the size of the dominant peak (note:the very low-frequency peak at around 0.001 Hz in figures 10 and 13 for a 400 p.p.m.fluid corresponds to a time-dependent viscosity trend occurring over nearly the entireperiod of 1024 s, which is much slower than the ‘spiral bursting’ secondary-flow cycle(as seen in video imaging and detected rheologically) that occurs at around 0.005 Hz.This low-frequency large-amplitude peak at around 0.001 Hz is not repeatable, as itdid not appear in a separate, identical experiment with a 400 p.p.m. fluid, as shown infigure 13). Finally, in the ‘elastic turbulent’ mode, the occasional loss of pattern centrewith device axis may not always occur (we have in some experiments observed this‘loss of centre’, and in other experiments, have either not observed it, or are uncertainif it occurred), although other important features of ‘elastic turbulence’ are present,including broad disordered spiralling and patterns which closely resemble the imagesof Groisman & Steinberg (2000, 2004); a dramatic increase in apparent viscosityof approximately 10 times the base flow viscosity for the 800, 400 and 200 p.p.m.solutions; and excitation of a broad spectrum of frequencies as seen in the FFT data.Also, for the 200 p.p.m. fluid in the ‘elastic turbulent’ mode, although the flow appearsturbulent based on image analysis and the broad spectrum of excited time-dependentmodes (see figures 8, 9 and 10), something resembling (or actually) ‘spiral bursts’

0 0.01 0.02

Frequency (Hz)

Figure 13. ‘Spiral bursting’ as a function of concentration. Plotted is frequency vs. power spec-trum, where frequency is an FFT of the time-dependent shear rate at fixed shear stress. Shown(from top to bottom) are the 800, 400 and 200 p.p.m. polyacrylamide (Mw = 5–6 × 106 g mol−1)fluids. For each concentration, results from two separate experiments are shown.

persists at approximately 0.014 Hz (as estimated from video imaging analysis), whichcorresponds to the peak in the FFT of the time-dependent shear rate at around0.016 Hz (see figure 10). We also recall here that another anomalous feature of the‘elastic turbulent’ mode for the 200 and 400 p.p.m. fluids is the marked decline inapparent viscosity (see figure 9); we speculated earlier that this may be due tothe possibility of shear-induced polymer migration, as Magda et al. (1993) reportedthe possibility of shear-induced polymer migration in similar viscoelastic fluids. It isunknown what effect the drop in apparent viscosity has on the ‘elastic turbulent’ mode.

0 0.02 0.04

0 0.02 0.040.06 0.06 0.06

0 0.02 0.04

0 0.02 0.040.06 0.06 0.06

0 0.02 0.04

0 0.02 0.040.06 0.06 0.06Pow

Frequency (Hz) Frequency (Hz) Frequency (Hz)

Figure 14. Progression from ‘spiral bursting’ to ‘elastic turbulence’ – example from a single800 p.p.m. polyacrylamide (Mw =5–6 × 106 g mol−1) fluid experiment. We show FFTs of thetime-dependent shear rate as a function of progressively increasing values of fixed shear stress,from ‘spiral bursting’ at 11.9 Pa to ‘elastic turbulence’ at 69.9 Pa and 84.8 Pa. The left-handcolumn (top to bottom) shows σ = 11.9 Pa, 13.4 Pa, 14.9 Pa. The middle column (top to bottom)shows σ = 19.9 Pa, 24.9 Pa, 29.9 Pa. The right-hand column (top to bottom) shows σ = 59.9 Pa,69.9 Pa, 84.8 Pa. Note the difference in y-axis scales (0.6 units for 11.9 Pa to 29.9 Pa, and 1.5units for 59.9 Pa to 84.8 Pa), and note that the very low-frequency point (at approximately0.001 Hz) for σ = 59.9 Pa is off scale, with a value of 3.0 units.

5.3. Additional observations

Thus far, we have presented evidence for the repeatable occurrence of distinct modes(in multiple fluids) that lead to ‘elastic turbulence’. The identified modes represent onlypart of what is seen in experiments lasting around 8 h. In general, the intermediatestates (between modes) appear to be a combination of both the secondary flowstructure and rheological signatures of the identified modes occurring before andafter these intermediate states. In figure 14, we show FFT of the time-dependentshear rates for the progression from ‘spiral bursting’ to ‘elastic turbulence’ thatoccurred in a single experiment for an 800 p.p.m. fluid at d/R = 0.204.

We include here additional observations of the ‘stationary ring’ mode fromexperiments using the molecular weight 5–6 × 106 g mol−1 polyacrylamide fluids. Inexperiments with 800 p.p.m. fluids, the ‘stationary ring’ mode shows a progressionfrom one bright ring to two concentric rings, indicating a likely increase in underlyingsecondary-flow amplitude and/or roll cell number. We have also observed thisprogression in a single 400 p.p.m. experiment, but are unable to assess if this multi-ringpattern is present in 200 p.p.m. fluid experiments because the image patterns are toofaint.

For the experiments shown in figure 6 using the 5–6 × 106 g mol−1 polyacrylamidefluids (each concentration was tested in duplicate), the range of Wi for the 800, 400 and

200 p.p.m. fluids was, respectively (here listed for both experiments): Wi= 0.20–9.1,0.088–11.2; 0.30–13.7, 0.16–16.9; 0.55–49.7, 0.31–48.0. For the same experiments, themaximum Re reached for the 800, 400 and 200 p.p.m. fluids was, respectively (listedhere for both experiments): 0.023, 0.028; 0.12, 0.14; 0.59, 0.57 (note: Re numbersreported here for the 200 p.p.m. fluid experiments are for shear stress values around100 Pa, when the fluid has already undergone or continues to undergo a major decreasein apparent viscosity, which results in a marked rise in shear rate and Re. When the‘elastic turbulent’ mode was reached for the 200 p.p.m. fluid shown in figures 8, 9 and10, at 59.6 Pa, Re was 0.23).

Finally, we have observed on repeated occasions (often after the onset of ‘elasticturbulence’ and then after the shear stress has been increased to a higher value)that the fluid can detach from part(s) of the cup edge region; all data points (i.e.shear stress steps) where this type of fluid detachment has been observed have beenexcluded.

6. DiscussionA striking feature of the cascade described above is that after a range of frequencies

is excited (see, for example, figure 5a), a single, dominant frequency stabilizes in thetime-dependent shear viscosity (near 0.006 Hz in this example; see figure 5b). Alsoimportant to the transition pathway is the evolving cyclical instability in the flow fieldcentre, which appears to connect ‘multi-spiral chaos’ to ‘spiral bursting’, leading to‘elastic turbulence’, and proceeds from temporal aperiodicity to periodicity and thenback to aperiodicity, respectively (note that in the case that ‘spiral oscillatory’ modeoccurs, the progression is from temporal periodicity to aperiodicity, to periodicity,and finally back to aperiodicity). We note additionally that the progression from onesecondary-flow mode to the next seems to occur without discontinuous rheologicalchanges for the most part.

The radial structure of the first secondary-flow mode (‘stationary ring’) is consistentwith the results of Byars et al. (1994). Furthermore, in our work, as stress increases,the inner ring only remains stable at progressively reduced radial positions (it movesinward), until it is ultimately replaced with spiral non-axisymmetric vortices at all ra-dial positions. Given that the local shear rate and hence the local Weissenberg numberincrease linearly with radial position in a torsional parallel-plate flow (for stable base-state flow), this suggests that the first instability in this experiment is an axisymmetricmode. Finally, our work resembles recent experiments on Newtonian flows at high Rein the torsional parallel-plate flow which show an analogous progression in symmetrybreaking of secondary-flow structures (Schouveiler et al. 2001).

We considered the possible unwanted effects of temperature gradients, inertia, andcapillary forces on the flow field. Using a thermocouple to measure fluid temperature,we confirmed the absence of viscous heating in these experiments. Examples ofmeasured temperature changes (the temperature before experimentation minus thetemperature just after the experiment has ended) for polyacrylamide test solutions(with very similar constituent concentrations to the 492 p.p.m. solution in this report)are +0.9, +0.1, −0.8 and −0.6 C, and we believe the fluid temperature closelyreflects room temperature. In addition to measuring fluid temperature, we estimated athermoelastic number for these experiments, Θ = Na1/2/De, where Na is the Nahme–Griffith number (Na = η0χ d2γ 2/kε, where χ = ε/η0(|dη/dε|), ε is the fluid’s tempera-ture in Kelvin units, and k is the fluid’s thermal conductivity) and De is the Deborahnumber for the torsional parallel-plate flow (De = λΩ , where Ω is disk angular

velocity; Rothstein & McKinley 2001). We estimate Θ to be around 0.002 in ourexperiments, suggesting that elasticity dominates over viscous heating, due in partto the large thermal conductivity of these aqueous solutions (note: in order toapproximate dη/dε, we used measurements made on an aqueous solvent system withsimilar sugar concentration to our actual solvent).

A control experiment using only solvent components (with no polymer additionand total solution viscosity comparable to the polymer solution) was performed withγ as the control parameter, and no patterns or pattern progression were seen likethose of the elastic instability modes. We note that at low shear rates, faint diffuseaxisymmetric patterns can form (which we believe to be due to colloidal forces), butthese disperse rather than intensify as shear rate increases, do not migrate radially,and are less distinct than the axisymmetric patterns of the elastic instability modes.Finally, rheological evidence of all elastic modes was observed in experiments withoutthe imaging-tracer particles, showing that the flow patterns (which have correspondingrheological signatures) were not themselves produced by the presence of the tracerparticles.

In order to provide a rough comparison between the stability of the elastic fluidsused in our experiments and the various elastic fluids used by previous workers tostudy elastic instabilities, we compare the critical Deborah numbers (De ≡ λΩ) of ourfluids in cone-and-plate experiments (see figure 2) to cone-and-plate data reportedby others (for a variety of cone angles and fluid compositions), for the first unstablemode (note that we only present here critical De when an Oldroyd-B relaxationtime for the fluid is available). In our experiments, the critical Deborah numberswere approximately 0.3, 0.5 and 0.6 for the 800, 400 and 200 p.p.m. polyacrylamidesolutions, respectively (as determined by rheological measurements). These criticalDeborah numbers are close to the low end of the range determined by Magda &Larson (1988) (approximately 1 to 9, as determined by rheological measurements,collectively, for the PIB fluids described above), and lower than the critical valuesdetermined by McKinley et al. (1991, 1995; De of 4.39, as determined by rheologicalmeasurements, and De of 4.95 to 7.60, as determined by secondary-flow imaging,respectively. The first set of reported measurements were for the 3100 p.p.m. PIBfluid described above, and the second set of measurements were for the high and lowcritical values for studies performed on the 3100 and 2000 p.p.m. PIB fluids describedabove relating to the work of Byars et al. 1994). We note that the relationshipbetween polymer concentration and β (in the range studied), and critical shear rateis evident in both the main experiments reported here (see figure 6b) as well as in thecone-and-plate measurements of the three fluid concentrations, which also showedinstabilities (see figure 2). Our results show that an increase in polymer concentrationand therefore in 1 − β correlates to earlier onset of instability as a function of shearrate (as well as shear stress in the case of figure 6a). Finally, we note that it was notpossible to align all the critical shear rates obtained in our experiments to a singlecritical condition by either plotting against Ψ1γ /η or by multiplying the shear rateby the relaxation time (i.e. plotting against Wi); we speculate that this may havesomething to do with our test solutions spanning the dilute to semi-dilute regimes.

Oscillatory (overstable) modes have been reported previously for purely elasticinstabilities, including those in torsional parallel plate flow experiments (Byars et al.1994) and in Taylor–Couette studies (Shaqfeh et al. 1992). In the parallel-plateexample, the oscillatory modes were detected by flow visualization and quantified bymeans of image analysis; as an example, the propagation frequency of a mode for the3100 p.p.m. fluid (described previously) at d/R = 0.05 was found to be 1.15 s (the fluid

relaxation time was 0.794 s). In a Taylor–Couette example, an oscillatory mode wasdetected by rheological measurement (a periodic oscillation in the measured torque, atconstant applied shear rate) for a high-molecular-weight polystyrene solution, with apropagation period of approximately 33 s (the fluid relaxation time was 1.27 s, wherethe relaxation time was an Oldroyd-B relaxation time). Despite the differences in flowgeometries and fluids used, we conclude that the time period of ‘spiral oscillatory’mode (approximately 260 s, as determined by FFT of the time-dependent torque) ismuch greater relative to the fluid relaxation time (2.3 s), compared to the examplestaken from these two previous studies (Shaqfeh et al. 1992; Byars et al. 1994).

The shear stress increment protocols used vary from experiment to experiment, but,in general, we attempted to explore multiple shear stress values in regions (of shearstress) where particular modes were expected to form, in experiments lasting around8 h. The modes we identified often became more distinct over a number of consecutiveshear stress steps, and then less distinct over a number of subsequent shear stresssteps (signs of subsequent modes also began to appear). For the experiments shownin this report, we have, for the most part, explored the first several modes with 10 minsteps (with 1 data point collected per second), and all other modes with 20 min longsteps (with 1 data point collected per second). The 20 min (1200 s) holds permittedus to collect enough time-dependent shear rate data for a 1024-point FFT (wherethe start-up effects could be eliminated by using only the final 1024 measurements tocalculate the FFT). The size of the shear stress increments for the experiments shownin this report can be seen in the spacing of the data points in figures 3 and 6(a).

For our study with 18 × 106 gmol−1 polyacrylamide, we acquired the same material(from the same commercial supplier) as first used by Groisman & Steinberg (2000).In our follow-up experiments, we used a 5–6 × 106 g mol−1 polyacrylamide material(obtained from a different commercial supplier), and this material behaved more elasti-cally than we would have expected, relative to our 18 × 106 g mol−1 polyacrylamidefluid. It turned out that our dilute (200 p.p.m.), borderline (400 p.p.m.) and semi-dilute(800 p.p.m.) concentrations of 5–6 × 106 g mol−1 polyacrylamide fluids all possessrelaxation times greater than our semi-dilute 18 × 106 g mol−1 fluid. Also, observedcritical shear rates for the 5–6 × 106 g mol−1 solutions versus the 18 × 106 g mol−1

solutions were not in line with what would be expected given the apparent differencein molecular weight. In addition, we did not find our 18 × 106 g mol−1 material to beas elastic as Groisman & Steinberg had observed (as indicated by the critical shearrate for onset of elastic instability and turbulence). Therefore, we conclude, that forwhatever reason, our 18 × 106 g mol−1 polyacrylamide may not be as high in molecularweight as advertised, though we are unable to readily characterize this material bymeans of gel permeation chromatography. Despite these discrepancies, we emphasizethat the modes originally discovered using the 18 × 106 g mol−1 polyacrylamide fluidwere verified using the 5–6 × 106 g mol−1 polyacrylamide fluids.

7. Conclusion and suggested future workIn summary, we have mapped a previously unexplored transitional pathway to tur-

bulence, discovering a rich series of secondary flow states connecting simple torsionalshearing flow in a parallel plate device to ‘elastic turbulence’, with negligible inertia.The secondary-flow states involve axisymmetric rolls and non-axisymmetric spiralsthat compete at high Weissenberg number, producing oscillations, apparent ‘chaotic’flow, and, eventually, apparent ‘turbulent’ flow with a broad spectrum of temporalfluctuations. It is often the case that experimental observations precede and inspire

new theoretical understanding, and these experimental observations provide newinsight into the nature of complex transitions and critical phenomena in elastic fluids.Much more is also to be learned from computational fluid mechanics and furtherexperimental studies of the transitions. Eventually, the results can be compared to thetransitions leading to turbulence in the corresponding Newtonian flows. In time, wehope that it will be possible to compare experimentally and theoretically the routesto purely elastic turbulence with those for inertial turbulence, leading to a richerunderstanding of both.

We thank former chemical engineering students J. Baxter, P. Desai and R.Cantor forcontributions in the laboratory, and J.Mears for advice and fabrication of parts of theflow-visualization device. We also thank T.A. Instruments’ A.Glasman, F.Mazzeo andC. Werzen for their assistance. B.A.S. was supported primarily by a US NIH/CBTP(Grant T32-GM08353) fellowship, and also by a US DOE/GAANN fellowship, underGAANN grant P200A0230-04. Support for other co-authors was provided by NASA,under grant NAG3-2708.

REFERENCES

Avagliano, A. & Phan-Thien, N. 1996 Torsional flow: elastic instability in a finite domain. J. FluidMech. 312, 279.

Avagliano, A. & Phan-Thien, N. 1999 Torsional flow stability of highly dilute polymer solutions.J. Non-Newtonian Fluid Mech. 84, 19.

Byars, J. A., Oztekin, A., Brown, R. A. & McKinley, G. H. 1994 Spiral instabilities in the flowof highly elastic fluids between rotating parallel disks. J. Fluid Mech. 271, 173.

Cros, A. & Le Gal, P. 2002 Spatiotemporal intermittency in the torsional Couette flow between arotating and stationary disk. Phys. Fluids 14, 3755.

Di Prima, R. C. & Swinney, H. L. 1985 In Hydrodynamic Instabilities and the Transition toTurbulence (ed. H. L. Swinney & J. P. Gollub). 2nd Edn. Springer.

Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53.

Groisman, A. & Steinberg, V. 2004 Elastic turbulence in curvilinear flows of polymer solutions.New J. Phys. 6, art. 29.

Jackson, K. P., Walters, K. & Williams, R. W. 1984 A rheometrical study of Boger fluids.J. Non-Newtonian Fluid Mech. 14, 173.

Landau, L. D. & Lifshitz, E. M. 1995 Fluid Mechanics, 2nd Edn. Butterworth Heinemann.

Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta. 31, 213.

Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.

Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely elastic instability in Taylor–Couetteflow. J. Fluid Mech. 218, 573.

McKinley, G. H., Byars, J. A., Brown, R. A. & Armstrong, R. C. 1991 Observations on theelastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid.J. Non-Newtonian Fluid Mech. 40, 201.

McKinley, G. H., Oztekin, A., Byars, J. A. & Brown, R. A. 1995 Self-similar spiral instabilitiesin elastic flows between a cone and a plate. J. Fluid Mech. 285, 123.

Magda, J. J. & Larson, R. G. 1988 A transition occurring in ideal elastic liquids during shear flow.J. Non-Newtonian Fluid Mech. 30, 1.

Magda, J. J., Lee, C. S., Muller, S. J. & Larson, R. G. 1993 Rheology, flow instabilities, andshear-induced diffusion in polystyrene solutions. Macromolecules 26, 1696.

Nguyen, H. & Boger, D. V. 1979 The kinematics and stability of die entry flows. J. Non-NewtonianFluid Mech. 5, 353.

Olagunju, D. O. 1994 Effect of free surface and inertia on viscoelastic parallel plate flow. J. Rheol.38, 151.

Phan-Thien, N. 1983 Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability.J. Non-Newtonian Fluid Mech. 13, 325.

Rothstein, J. P. & McKinley, G. H. 2001 Non-isothermal modification of purely elastic flowinstabilities in torsional flows of polymeric fluids. Phys. Fluids 13, 382.

Schouveiler, L., Le Gal, P. & Chauve, M. P. 2001 Instabilities of flow between a rotating and astationary disk. J. Fluid Mech. 443, 329.

Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28,129.

Shaqfeh, E. S. G., Muller, S. J. & Larson, R. G. 1992 The effects of gap width and dilute solutionproperties on the viscoelastic Taylor–Couette instability. J. Fluid Mech. 235, 285.

Swinney, H. L. & Gollub, J. P. 1985 Hydrodynamic Instabilities and the Transition to Turbulence,2nd Edn. Springer.

Weissenberg, K. 1947 A continuum theory of rheological phenomena. Nature 159, 310.

Transitional pathway to elastic turbulence in torsional, parallel-plate flow of a polymer solution

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